05C38 Paths and cycles [See also 90B10]
Dynamic network flow problems model the temporal evolution of flows over time and also consider changes of network parameters such as capacities, costs, supplies, and demands over time. These problems have been extensively studied in the past because of their important role in real world applications such as transport, traffic, and logistics. This has led to many results, but the more challenging continuous time model still lacks some of the key features such as network related optimality conditions and algorithms that are available in the static case.
The aim of this paper is to advance the state of the art for dynamic network flows by developing the continuous time analogues of several well-known optimality conditions for static network flows. Specifically, we establish a reduced cost optimality condition, a negative cycle optimality condition, and a strong duality result for a very general class of dynamic
network flows. The underlying idea is to construct a dual feasible solution that proves optimality when the residual network (with respect to a given flow) contains no dynamic cycles with negative cost. We also discuss a generic negative cycle-canceling algorithm resulting from the corresponding optimality criterion and point out promising directions for future research.
Based on a recent work by Abraham, Bartal and Neiman (2007), we construct a strictly fundamental cycle basis of length O(n2) for any unweighted graph, whence proving the conjecture of Deo et al. (1982).
For weighted graphs, we construct cycle bases of length O(W log(n) log(log(n))), where W denotes the sum of the weights of the edges. This improves the upper bound that follows from the result of Elkin et al. (2005) by a logarithmic factor and, for comparison from below, some natural classes of large girth graphs are known to exhibit minimum cycle bases of length Ω(W log(n)).
We achieve this bound for weighted graphs by not restricting ourselves to strictly fundamental cycle bases - as it is inherent to the approach of Elkin et al. - but rather also considering weakly fundamental cycle bases in our construction. This way we profit from some nice properties of Hierarchically Well-Separated Trees that were introduced by Bartal (1998).